Pendulum Neuron: Oscillatory Spiking Model

• Pendulum neurons are a neuron model where the state is represented by a damped, driven oscillator with phase dynamics that trigger spikes through threshold crossings.
• The model leverages wave-to-pulse conversion, synchronization, and interference techniques to encode information beyond traditional integrative approaches.
• Its framework supports neuromorphic computing and adaptive network organization, offering novel insights into phase-based and resonator neuron dynamics.

Searching arXiv for relevant papers on pendulum neuron and closely related oscillator-based neuron models. {"query":"all:(\"pendulum neuron\" OR \"Pendulum Model of Spiking Neurons\" OR \"Wave to pulse generation\" OR \"resonator neuron pendulum\" OR \"Purkinje neuron oscillator\" OR \"synchronized pairs of nano-oscillators\")","max_results":10,"sort_by":"relevance"} A pendulum neuron is a neuron model, or neuron-like computational element, whose internal state is represented as an oscillator with pendulum-like dynamics rather than as a purely first-order leaky integrator. Across the literature, the term encompasses several related constructions: synapses and neurons treated as oscillatory systems that smooth spike trains into waves and recover spikes through thresholding (Castellanos, 2018); explicitly second-order spiking neurons governed by a damped, driven pendulum equation with phase-threshold reset (Bose, 29 Jul 2025); phase-oscillator and coupled-oscillator implementations in physical substrates such as nano-oscillators and lasers (Vodenicarevic et al., 2017, Dolcemascolo et al., 2018); and biologically grounded oscillator interpretations of neural dynamics, including Purkinje neurons modeled as tunable parametric oscillators (Abrams et al., 2011). The common thread is that information is carried not only by spike count or membrane amplitude, but by phase, frequency, resonance, synchronization, and interference, with spikes emerging as thresholded events from an underlying oscillatory state.

1. Conceptual scope and definition

In the most direct formulation, a pendulum neuron is a spiking neuron whose internal phase evolves according to a damped, driven pendulum rather than a leaky integrator. In "Pendulum Model of Spiking Neurons" (Bose, 29 Jul 2025), the neuron’s state is an angular phase $\theta(t)$, and the model is designed to incorporate oscillatory behavior, damping, nonlinear phase dynamics, and input-driven timing modulation. A spike is emitted when the phase reaches a threshold and the state is reset, so spike timing is explicitly a phase-crossing phenomenon rather than a scalar voltage-threshold event.

A broader usage appears in work that treats synapses or neurons as oscillatory systems with phase, amplitude, frequency, and coupling. "Wave to pulse generation. From oscillatory synapse to train of action potentials" (Castellanos, 2018) models the synapse as an oscillatory system that transforms a presynaptic spike train into a smoothed wave and then, through collective interference and thresholding, back into action potentials. This suggests a generalized pendulum-neuron viewpoint in which the operative variable need not be a literal angle, but an oscillator state with conserved temporal structure.

A related but distinct line of work uses pendulum-like or coupled-oscillator systems as computational neurons in neuromorphic or machine-learning settings. A pair of synchronized nano-oscillators can implement a neuron through their phase-locked response curve (Vodenicarevic et al., 2017), while a single driven pendulum can act as a reservoir in reservoir computing (Mandal et al., 2022). These systems are not always posed as biological neurons, but they instantiate the same principle: nonlinear oscillatory physics provides the neuron’s transfer function, memory, and timing sensitivity.

2. Canonical dynamical formulations

The explicit pendulum-neuron equation proposed for spiking neurons is

$$\frac{d^2\theta}{dt^2} + \gamma \frac{d\theta}{dt} + \omega^2 \sin(\theta) = I(t),$$

where $\theta(t)$ is the angular phase, $\gamma$ is the damping coefficient, $\omega$ is the natural frequency, and $I(t)$ is the external input current or torque (Bose, 29 Jul 2025). The corresponding first-order form used for simulation and Brian2 implementation is

$\frac{d\theta}{dt} = \omega,$

$$\frac{d\omega}{dt} = -\gamma \omega - \omega_0^2 \sin(\theta) + I.$$

In that model, the spike rule is a phase threshold:

• threshold: theta > pi
• reset: theta = 0; omega = 0 (Bose, 29 Jul 2025)

This construction makes the neuron a second-order nonlinear oscillator with explicit phase encoding. By contrast, the earlier wheel model used only

$\frac{d\theta}{dt} = \omega$

or, with input,

$\frac{d\theta}{dt} = \omega + a I(t),$

and therefore lacked inertia and damping (Bose, 29 Jul 2025).

A broader oscillator-based abstraction appears in "Wave to pulse generation" (Castellanos, 2018), where the synapse is modeled as a linear time-invariant system in the subthreshold regime. Neurotransmitter diffusion in the cleft is described by

$$\frac{d^2\theta}{dt^2} + \gamma \frac{d\theta}{dt} + \omega^2 \sin(\theta) = I(t),$$0

with Fourier-space solution

$$\frac{d^2\theta}{dt^2} + \gamma \frac{d\theta}{dt} + \omega^2 \sin(\theta) = I(t),$$1

and spatial Gaussian convolution

$$\frac{d^2\theta}{dt^2} + \gamma \frac{d\theta}{dt} + \omega^2 \sin(\theta) = I(t),$$2

Under the assumed proportionality $$\frac{d^2\theta}{dt^2} + \gamma \frac{d\theta}{dt} + \omega^2 \sin(\theta) = I(t),$$3, the synaptic transformation becomes a time-domain Gaussian convolution of the input pulse train. For a periodic Dirac-comb input,

$$\frac{d^2\theta}{dt^2} + \gamma \frac{d\theta}{dt} + \omega^2 \sin(\theta) = I(t),$$4

the output is a periodic train of Gaussians with the same period $$\frac{d^2\theta}{dt^2} + \gamma \frac{d\theta}{dt} + \omega^2 \sin(\theta) = I(t),$$5, implying preservation of the fundamental frequency (Castellanos, 2018).

Other pendulum-like neuron constructions retain oscillator dynamics but use different state spaces. In "Resonator neuron and triggering multipulse excitability in laser with injected signal" (Dolcemascolo et al., 2018), the reduced phase dynamics follow the Adler equation,

$$\frac{d^2\theta}{dt^2} + \gamma \frac{d\theta}{dt} + \omega^2 \sin(\theta) = I(t),$$6

which the paper interprets as an overdamped pendulum with fluid torque. To capture resonator behavior, an inertial extension is introduced:

$$\frac{d^2\theta}{dt^2} + \gamma \frac{d\theta}{dt} + \omega^2 \sin(\theta) = I(t),$$7

which is explicitly a driven pendulum with viscous damping (Dolcemascolo et al., 2018).

3. Oscillatory coding, wave-to-pulse transformation, and interference

A central feature of pendulum-neuron models is that they place oscillatory dynamics prior to spike generation. In the synaptic wave model, presynaptic spikes are smoothed by diffusion into a continuous wave, and periodic structure is preserved because convolution with a time-stationary kernel does not change the period of a periodic input (Castellanos, 2018). The output of the synapse is therefore a train of overlapping Gaussians with the original spike frequency, which the paper interprets as a smooth oscillatory field.

The next step is wave-to-pulse conversion. The collective dendritic input is written as a linear superposition,

$$\frac{d^2\theta}{dt^2} + \gamma \frac{d\theta}{dt} + \omega^2 \sin(\theta) = I(t),$$8

or, in the sinusoidal idealization,

$$\frac{d^2\theta}{dt^2} + \gamma \frac{d\theta}{dt} + \omega^2 \sin(\theta) = I(t),$$9

so constructive and destructive interference depend on the relative phases $\theta(t)$0 (Castellanos, 2018). Spikes arise when this summed oscillatory signal exceeds threshold. The paper does not derive a detailed Hodgkin–Huxley or integrate-and-fire spiking equation, but it explicitly frames spike emission as a nonlinear thresholding of collective oscillatory input, with an absolute refractory period of about 1 ms and a later relative refractory period (Castellanos, 2018).

This oscillatory interpretation is closely aligned with the pendulum-neuron formulation of (Bose, 29 Jul 2025), where input alters the phase trajectory of a nonlinear oscillator, and spikes are emitted at particular phases. In both cases, the informative variables are timing, frequency, phase alignment, and resonance rather than spike shape. "Wave to pulse generation" further states that neurons are subthreshold approximately 99 percent of the time, so most information processing is attributed to small signals in the subthreshold regime rather than to action potentials themselves (Castellanos, 2018).

The same coding logic appears in the laser-based resonator neuron. There, external perturbations act as kicks to a phase oscillator. If the perturbation crosses the separatrix, the phase executes a $\theta(t)$1 rotation, which is interpreted as a spike; larger perturbations can induce multiple rotations and therefore multiple pulses (Dolcemascolo et al., 2018). This suggests a common pendulum-neuron template: a stable oscillatory or near-oscillatory state, a threshold or separatrix, and discrete spike events corresponding to large excursions in phase space.

4. Resonance, synchronization, and phase-based computation

Pendulum-neuron models differ from first-order integrate-and-fire descriptions by embedding natural oscillation and resonance into the state dynamics. "Pendulum Model of Spiking Neurons" argues that LIF neurons do not naturally express oscillations, resonance, or phase coding, whereas the pendulum neuron is designed precisely to incorporate those properties (Bose, 29 Jul 2025). With constant input, the model produces periodic spiking whose frequency depends on $\theta(t)$2 and input intensity $\theta(t)$3; with oscillatory input such as

$\theta(t)$4

spike times become phase-locked to the input rhythm (Bose, 29 Jul 2025). The paper also notes that pulsed inputs induce phase-response-curve–like behavior, since the effect of a pulse depends on the current phase.

The resonator–integrator distinction is especially explicit in the injected-laser system. In the Adler limit, the system behaves as a Class I integrator: two subthreshold perturbations are effective mainly when close in time, and spike probability increases monotonically as delay decreases (Dolcemascolo et al., 2018). Outside that approximation, with effective inertia and relaxation oscillations, the system behaves as a resonator: efficiency peaks at specific interpulse delays because the second perturbation is most effective when matched to the phase of damped oscillations (Dolcemascolo et al., 2018). This is the pendulum analogy in a strict dynamical-systems sense: an underdamped oscillator exhibits timing-sensitive forcing, whereas an overdamped one primarily sums inputs.

Synchronization is the network-level extension of the same idea. "Wave to pulse generation" explicitly relates synapses and neurons to coupled nonlinear oscillators and cites synchronization results associated with Van der Pol, Katchalsky, and Strogatz (Castellanos, 2018). In that view, weakly coupled oscillator units can phase-lock, and synchronized constructive interference increases the probability of threshold crossing. "A Neural Network Based on Synchronized Pairs of Nano-Oscillators" operationalizes this in hardware-like form: a neuron is built from two bidirectionally coupled oscillators governed by

$\theta(t)$5

with one oscillator’s natural frequency modulated by the weighted input sum (Vodenicarevic et al., 2017). In the synchronized regime, the phase difference encodes the input-dependent detuning, and the output is extracted from the envelope amplitude of the summed signals.

The resulting transfer function is non-monotonic:

$\theta(t)$6

with normalized output

$\theta(t)$7

defined inside the synchronization region $\theta(t)$8 (Vodenicarevic et al., 2017). Because of that unconventional response curve, a single oscillator neuron can implement XOR and XNOR, which a single sigmoid or threshold neuron cannot (Vodenicarevic et al., 2017).

5. Biological grounding and network organization

Although some pendulum-neuron models are explicitly artificial, others are motivated by biological oscillator dynamics. "Tunable Oscillations in the Purkinje Neuron" treats Purkinje neurons as tunable parametric oscillators and derives a second-order equation for membrane voltage from a Hodgkin–Huxley framework (Abrams et al., 2011). Starting from

$\theta(t)$9

the paper differentiates and separates slow from fast terms to obtain a reduced second-order form compared directly with

$\gamma$0

with the identifications

$\gamma$1

Because the effective damping and stiffness depend on gating variables, the system is a forced parametric oscillator rather than a fixed-coefficient linear oscillator (Abrams et al., 2011). Experimentally, slow oscillations are induced in every Purkinje neuron tested, with periods ranging between 10 and 25 seconds, and the neurons return to their intrinsic firing frequency after forced oscillation is concluded (Abrams et al., 2011). This is a literal biological realization of a pendulum-like neuron in the sense of second-order oscillatory dynamics with tunable forcing and recovery to an intrinsic frequency.

The network extension of pendulum neurons is developed explicitly in (Bose, 29 Jul 2025). Each neuron carries its own phase $\gamma$2 and angular velocity $\gamma$3, while synaptic weights $\gamma$4 are updated by Hebbian and STDP rules. The STDP update is given as

$\gamma$5

and instantiated in pseudocode as

$\gamma$6

$\gamma$7

with $\gamma$8 (Bose, 29 Jul 2025). Because spike times are determined by continuous phase evolution, STDP operates not merely on event order but on oscillator phase relationships. The paper states that such neurons form oscillatory phase-locked patterns and can learn temporal associations in symbolic sequences, such as character streams or rhythmic patterns (Bose, 29 Jul 2025).

A more abstract adaptive pendulum network appears in "Emergence of solitary and chimera states in adaptive pendulum networks under diverse learning rules" (Anand et al., 11 Mar 2026). There each unit obeys

$\gamma$9

with adaptive couplings

$\omega$0

For Hebbian adaptation, $\omega$1 yields $\omega$2; for STDP, $\omega$3 yields $\omega$4 (Anand et al., 11 Mar 2026). Under Hebbian adaptation the network exhibits two-cluster, solitary, multi-antipodal, and chimera states, while STDP produces splay, splay-cluster, and splay-chimera configurations (Anand et al., 11 Mar 2026). This shows that pendulum-like neuron networks can support multistable collective states usually associated with neural synchronization theory.

6. Computational implementations and neuromorphic control

Pendulum-neuron ideas have also been developed as computational substrates and control architectures. "Machine Learning Potential of a Single Pendulum" uses a single driven pendulum as a reservoir for reservoir computing, with rich transient dynamics providing the computational resource (Mandal et al., 2022). The core idea is that even a low-dimensional nonlinear system can serve as a suitable candidate for a reservoir by exploiting transient dynamics, and the paper states that this single simple nonlinear system can successfully perform temporal and non-temporal tasks (Mandal et al., 2022). This suggests a pendulum neuron in the reservoir-computing sense: one nonlinear oscillator, sampled in time, furnishes a virtual high-dimensional state.

Event-based pendulum control provides a reciprocal interpretation in which the controller itself is neuron-like. "Neuromorphic Control of a Pendulum" models both pendulum and controller as event-based systems that coordinate their rhythms through properly timed events (Schmetterling et al., 2024). The plant is

$\omega$5

with actuation delivered as short torque pulses generated by neuromorphic half-center oscillators (Schmetterling et al., 2024). The controller uses photodetector events at angle crossings, phase response curves, and adaptive modulation of burst size and frequency. The paper treats the control problem as mixed rhythmic automaton design plus feedback timing regulation, and the architecture amounts to a spiking neural rhythm generator phase-locking to a mechanical oscillator (Schmetterling et al., 2024).

"A hybrid systems analysis" formalizes a related setup in which spikes are modeled as Dirac delta torque pulses applied whenever the pendulum crosses its resting position (Petri et al., 8 Apr 2025). The linearized pendulum dynamics are

$\omega$6

and jumps are defined by

$\omega$7

The paper proves the existence, uniqueness, and uniform exponential stability of a hybrid limit cycle for the closed-loop system (Petri et al., 8 Apr 2025). A plausible implication is that pendulum-neuron architectures can be analyzed not only heuristically but also with rigorous hybrid systems tools when the spikes are modeled as impulsive events.

A further control-oriented realization is provided by deterministic spiking networks that learn feedback control for the cart-pole or inverted pendulum (Kang et al., 2017). There the plant-controller loop is a hybrid dynamical system, the controller output is determined by precise spike times, and the learned stability region is comparable to that of a PID controller (Kang et al., 2017). The results are not framed as a pendulum neuron internally governed by a pendulum equation, but they reinforce the larger theme that spike timing and event-based dynamics can control pendulum systems efficiently.

7. Comparisons, limitations, and open questions

Several comparisons recur across the literature. Relative to LIF, the pendulum neuron in (Bose, 29 Jul 2025) is second-order, explicitly nonlinear, and phase-encoding; relative to the Izhikevich model, it lacks an adaptation variable but provides direct phase-based dynamics and complex oscillatory computation (Bose, 29 Jul 2025). Relative to the Adler or $\omega$8-neuron reduction, the full laser model in (Dolcemascolo et al., 2018) captures resonator behavior and multipulse excitability that the overdamped reduction cannot. Relative to standard feedforward neural nonlinearities, the synchronized nano-oscillator neuron offers a non-monotonic transfer curve with native XOR capability (Vodenicarevic et al., 2017).

The limitations are equally consistent. "Pendulum Model of Spiking Neurons" notes computational cost, parameter sensitivity, lack of explicit adaptation, and limited empirical benchmarking (Bose, 29 Jul 2025). "Wave to pulse generation" is conceptual and assumes 1D diffusion, homogeneous cleft structure, constant neurotransmitter velocity, and no fitted biophysical parameters; it also notes limited direct biological evidence for cleft-scale ion and neurotransmitter dynamics (Castellanos, 2018). The Purkinje-neuron oscillator model is tied to slow in vitro oscillations and does not establish that all neural timing phenomena are pendulum-like (Abrams et al., 2011). Hardware-oriented oscillator neurons remain constrained by synchronization regions, device variability, and on-chip learning challenges (Vodenicarevic et al., 2017).

A common misconception is that a pendulum neuron is merely a metaphor for periodic firing. The cited work shows a narrower and more technical meaning. In some cases, the governing equation is literally that of a damped, driven pendulum (Bose, 29 Jul 2025, Dolcemascolo et al., 2018). In others, the system is mathematically equivalent to a parametric oscillator or to coupled phase oscillators with inertia and damping (Abrams et al., 2011, Anand et al., 11 Mar 2026). The defining feature is not generic periodicity but the use of oscillator state variables—phase, angular velocity, interference amplitude, or synchronization manifold—as the substrate of neural computation or spike generation.

Another misconception is that pendulum neurons are exclusively biological models. The literature spans biological interpretation, artificial neuromorphic design, and physical-computing realizations. The same oscillator formalism appears in spiking-neuron models (Bose, 29 Jul 2025), synaptic wave models (Castellanos, 2018), reservoir computing (Mandal et al., 2022), nano-oscillator neural hardware (Vodenicarevic et al., 2017), optical excitable systems (Dolcemascolo et al., 2018), and adaptive oscillator networks with Hebbian or STDP-like coupling (Anand et al., 11 Mar 2026). This suggests that "pendulum neuron" names a family of oscillator-based neural abstractions rather than a single canonical model.

Across these works, the most stable synthesis is that a pendulum neuron is a neuron or neuron-like unit whose state evolution is organized by oscillator mechanics: damping, forcing, phase advance, resonance, synchronization, and threshold crossings. In that framework, spikes are emergent discrete events riding on a continuous oscillatory substrate, and computation is naturally expressed in phase relations, interference structure, and adaptive coupling rather than in scalar membrane integration alone.